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* Corresponding author, E-mail: mad@cea.fr
INTRODUCTION
MODELLING: DEFINITION AND PRINCIPLES Static versus dynamic Spatial (usually 2-d or 3-d) versus local or non-spatial (0-d) Linear versus nonlinear Discrete versus continuous Deterministic versus probabilistic (stochastic) Top-down versus bottom-up MODELLING: TOOLS MODELLING: METHODS Ask the right question Choose the right tools Analyse the results STARTING KIT: SUGGESTED READING CONCLUSION: GOALS REFERENCES ## INTRODUCTIONNowadays, physics is often seen as the science of the inanimate world, from the very small (particle physics, quantum mechanics), to the very large (astronomy and cosmology), but this is only a recent restriction. Etymologically, physics comes from the Greek verb which means I appear and grow spontaneously like a cell or a plant. From this verb comes the substantive , which means nature and the related adjective . Hence, physics, or is the science of nature, and its object is the totality of the real world. Beyond, begins the realm of metaphysics. It is the reductionist approach which first permitted some fields of science to obtain results such as laws and equations: astronomy became a predictive science when planets were ‘reduced’ to points in the equations of Kepler and Newton. Because the sciences dealing with life could not for a long time benefit from this kind of approach, they were left out of mainstream physics. But recent progress in three very different directions – firstly, measuring methods and data analysis, secondly, theoretical tools, and lastly, computers – have drastically changed the picture, and now life sciences can be addressed by the tools of physics. Experimental physics has given a large number of
new tools to study nature of which the most notable are
microscopy, On the theoretical side, Friedrich Hayek, (1899–1992, Nobel prize in economics in 1974), was one of the first to introduce the concept of complexity in science, and he distinguished the possibility of predicting the behaviour of a simple system by using a law (an equation relating a small number of variables), from the possibility of predicting the behaviour of a complex system by using a model, and this applied to such fields as economy, biology, ecology, and psychology. Note that although there is no strict definition of a complex system, there is some consensus that it should: (a) have many components (b) have a behaviour which is not trivial to predict (c) exhibit some kind of emerging properties such as self-organization. Because of the large (and often huge) number of actors and relations in such systems, it is impossible to find simple laws describing their behaviour, and a very large number of equations and variables are necessary, thus constituting a model. This large number of equations cannot be solved by hand, and this is where the third element is crucial: the appearance of computers renders feasible the study of the evolution of the sets of equations which constitute the models. Modelling complex systems such as those encountered in meteorology, climatology, economy, social sciences, neurosciences, organic chemistry, molecular biology, psychology, and ecology is now possible using the methods of physics and supercomputers. Surprisingly (but is it really a coincidence?), it is at this same present time that direct human influence on our planet is becoming worrying. Short term problems have to be addressed urgently if we want to maintain the Earth as viable for mankind. And for this, it is no longer enough to discover, describe, and classify new species; one needs a real understanding of climate change, of ecosystems dynamics, of emerging diseases, and so on, in order to predict the impacts of human activities, and to adapt and optimize them. Fig. 1 shows a few spatio-temporal scales of living
systems (here, forest) and of meteorological phenomena,
while the two diamonds show the spatio-temporal
influence range of pre-industrial and modern man,
respectively. This kind of representation (without
man) was introduced in 1986 by C.S. Holling To address those problems, scientists working in life sciences have to build models of the objects they study, and they have to follow the paths and methods which have previously been used by physicists to model and understand inanimate objects. As will be discussed later, mathematics is but a tool to achieve this goal. This article does not intend to provide the reader with a complete understanding or an exhaustive inventory of existing modelling techniques for natural sciences, but simply to identify the goals and tools of modelling, and also its pitfalls. Above all, it attempts to convince the reader that work in collaboration with a physicist or a mathematician to build a model will offer rewards in the form of a better understanding. Modelling nature is not exclusive to the physicist or mathematician: it is the result of a team effort, where natural scientists are major players. Not only are they the end-users of an abstract product, but they are also essential in its making. ## MODELLING: DEFINITION AND PRINCIPLESThe definitions of ‘model’ (noun) in the Compact Oxford
English Dictionary are as follows: Science always follows a simple three-step process: (1) observation of a real phenomenon, e.g., the motion of planets, a tornado, the spread of a disease, the growth of a tree (2) elaboration of an abstract construction to represent or explain the observation; as seen earlier, it can be an equation, then called a law (like Newton’s law), a theory (which is basically a set of laws), or a model (3) comparison of the results or predictions of this construction with observation. If there is a strong disagreement, then go back to the second step. Two points should be made here. First, the criterion of empirical falsifiability, introduced by Karl Popper, is essential to the scientific method in physics and many other sciences. A theory or a model cannot be ‘proven’ but only ‘refuted’ for logical inconsistency, or shown wrong by experimentation. A ‘good’ theory or model is one which withstands confrontations with experimentation. On the other hand, a conjecture in mathematics, which does not deal with physical objects, can be refuted (by a counter example) but can also be proven (and hence become a theorem). But a theory or a model can be wrong (as everybody knows, general relativity has replaced Newton’s theory of gravity), but can still be useful (planet orbits can be described with a good degree of accuracy using classical mechanics). Secondly, it is very important to make the distinction between prediction and understanding. A model can be purely descriptive. Models can have good predicting abilities, without incorporating underlying mechanisms – they are like a ‘black box’ into which one feeds input quantities, such as initial conditions, and which produce predicted quantities as output. Conversely, an explicative model is built from causal mechanisms and can therefore be termed a theory. Models which are oriented essentially towards prediction (often, statistical models) are different from those that are built mainly to help to understand a phenomenon (generally, mechanistic models). Nevertheless, once a good mechanistic model is built, it can also become useful for prediction. Some other dichotomies should be defined, as listed below: ## Static versus dynamicA static model does not involve time, while a dynamic model does. ## Spatial (usually 2-d or 3-d) versus local or non-spatial (0-d)If the model is homogeneous (the system has the same state throughout), the variables are lumped, whereas if the model is heterogeneous (varying state within the system), then the variables are distributed or space-dependent. ## Linear versus nonlinearIf all the equations in the model are linear equations, the model is known as a linear model. If one or more of the objective functions or constraints are represented by nonlinear equations, then the model is a nonlinear model. ## Discrete versus continuousIn real life, one tends to consider that space, as well as time, are continuous. The phenomenon is then described by sets of either ordinary or partial differential equations (ODEs or PDEs). In many cases, however, models will consider space and/or time as discrete: time will proceed by jumps (which can be very short – fractions of seconds, or long – a year or more), while a surface will be described as a grid of squares and the variables will be computed on these squares. If the variables themselves are continuous, one speaks of finite difference equations (FDEs) (discrete time), or of coupled networks (discrete space and time). If the variables themselves take only discrete values, one will speak of cellular automata. ## Deterministic versus probabilistic (stochastic)A fully deterministic model performs exactly in the same way for a given set of initial conditions, while in a stochastic model, randomness is present and even when given an identical set of initial conditions, results may in some cases vary greatly (this would be the case for the outbreak of an epidemic started by one infected mosquito – if it dies before biting a host, there will be no incidence of the disease, whereas in another run of the model, the disease could reach the entire population). In other cases, the results have a limited dispersion, at least for averaged output variables (e.g., the epidemiological status of a given individual may vary from run to run, but the percentage of infected hosts would stay the same). The model is then said to be robust. It is to be noted that mechanistic models can incorporate some randomness (for instance in the expression of transition rates between different states). They are then no longer fully deterministic, but if care is taken to verify the robustness, they can still be considered as explicative and mechanistic. ## Top-down versus bottom-upIn many cases an idea or a theoretical model has been
conceived before envisaging a specific application to a
natural object. Reaction-diffusion equations and cellular
automata were studied long before they were applied to
morphogenesis of the patterns observed on shells and
animal pelts (see, e.g., Ref. 5) or “tiger bush” in Niger But often the problem comes first, and there exists no ready-made model or theory. The choice of modelling tools has to be decided after the question has been precisely formulated. It is then a “bottom-up” approach, from a practical problem towards an abstract formalization. Of course, once the modelling goal is reached, the physicist can try to find regularities and generalizations which can be compared to what is observed in more traditional physics. ## MODELLING: TOOLSA good modeller should be familiar with a number of concepts and methods. If one makes a short – and far from exhaustive – list, it should at least include the following: Hamiltonian theory, integrability, chaos, ergodicity, KAM tori, Lyapunov exponents, separatrices, jacobians, bifurcation theory, dissipative systems, (strange) attractors, fractals, percolation, solitons, deterministic models, stochastic models, probabilistic models, ODEs, PDEs, perturbation theory, FDEs, coupled lattices, cellular automata, wavelet analysis, neural networks, and Monte Carlo methods. The reader should not be frightened by this list: it is only a toolbox. Modelling is the work of a team, and the most important thing is not the equations or the computer programs, but the exchange of information and ideas between the different specialists (ecologists, biologists, geologists, epidemiologists, etc. and physicists). It is only after constructive interactions between all of these specialists that the modeller (physicist or mathematician by training) can choose from their toolbox the relevant tools to make a model of the problem being addressed. The worst way to make a model is to work with somebody who is a specialist in one and only one of these mathematical tools, and who will not try to adapt their toolbox to the problem, but will bend reality to be compatible with the tool they know well. Again, a model is built at the crossroads of different scientific branches and the modeller is somebody who listens and catalyses the formation of ideas, relations, datasets and hypotheses, before writing equations. In addition, a scientific model is something that is never finished, as it has to be put into question again whenever new facts either contradict it or cannot readily be incorporated into it. Many models have been proposed in recent years.
They are often based on compartmental analyses and use a
very wide range of methods from physics, mathematics,
or statistics. It is important to know what one wants
exactly before selecting which drawer of the toolbox to
open first. Differential equations, coupled lattices, cellular
automatons, stochastic processes, statistical estimations,
non-parametric or semi-parametric methods, artificial
neural networks analysis or sophisticated Markov chain
Monte Carlo methods can be used in epidemiological or
ecological modelling, but they will not all answer the
same questions. There is no universal method to model
complex systems in nature. The problem itself will lead to
the choice of the method. For instance, multivariable
statistical analysis is by itself a descriptive model
from which one can infer underlying mechanisms
and relations. Combined with multiple regression, it
can be used to build predictive models which will be
operational in the range of parameters covered by
the data they use. But recent examples, such as the
shameful dispersion in the predictions of BSE (bovine spongiform encephalopathy, or “mad cow disease”)
epidemics of more than three orders of magnitude
show the limits of such methods, which are ill adapted
to extrapolate. Artificial neural networks can also
be used for prediction ## MODELLING: METHODSFrom now on, we will talk only of explicative (mechanistic) models. To begin with, we can distinguish three steps in the mechanistic modelling of a natural system or process. ## Ask the right questionIn most cases, a global model of the object studied is not
needed. Often, the question is down to earth, and the goals
are urgent and practical: how to understand and protect a
given type of ecosystem, how to save what can be saved,
and otherwise, how to use its space and the life on it in an
optimal way, and this at a given place and a given time.
Confronted with such a well defined question which might
seem simple but which in fact is often extremely complex,
the physicist has to collaborate with the specialists of the
other disciplines to determine what are the essential
factors and variables, and what are the relations between
them. In the case of the protection of a local ecosystem for
instance, ecologists, botanists, and geologists will have
to collaborate with economists, geographers, and
sociologists, and the modelling processes starts with
the confrontation of these different points of view.
Only once a consensus has been reached on what
are the essential ingredients can the choice of the
mathematical tools be addressed. It is the same process in
epidemiology: take the case of Rift Valley fever, first
described in Eastern Africa in 1927 by Daubney et al ## Choose the right toolsMechanistic modelling has understanding as the
primary goal. In French, “to understand” translates
as “comprendre”, from the Latin cum-prehendere
which means “to take together”. It is in essence a
‘complex systems’ approach, which puts together –
as a mechanistic model – the hypotheses about the
elementary processes. The mathematical tools are
chosen from a large toolbox. Sometimes a suitable
approach is not present, and new theoretical tools have
to be developed specifically to address a new class
of problems, e.g., small-world models One could ask “what is best?” It is not a good
question, because there is no unique answer. The method
should be adapted to the problem being dealt with. To
choose a method to model a real natural phenomenon, be
it in ecology, biology, or epidemiology, the first thing is to
properly define the real questions asked, and then to make
an inventory of available data. If a modeller says they
need several thousands of non-measurable parameters to
build a model, they will get nowhere! Nevertheless, in
some cases, additional data will have to be collected One should start with as minimal a model as possible (“small is beautiful”), and then develop it and compare its output with experimental data. The systematic exploration of the model parameter space will give an understanding of the behaviour of the system, and it will then be possible to compare its output with reality. It is only if the model is unsatisfactory that additional components should be added. ## Analyse the resultsVery often, the first model fails to reproduce the behaviour
of the system studied. It is then necessary to reconsider
the important factors, the mechanisms which link
them, and to improve the model, which sometimes
means that other factors and mechanisms have to be
included. A good example is found in several problems in
epidemiology, which cannot be modelled using a 0-d approach, and for which it has been shown that 2-d
geometry has to be used (see, e.g., Refs. 13, 20). When
a satisfactory agreement between observation and
model output has been reached, it is then possible to
make predictions and to study the effects of different
strategies. Generally, one can observe the emergence of
self-organization, which is a characteristic of complex
systems and is the key to life, from the cell to the whole
planet (Lovelock’s “Gaia” concept ## STARTING KIT: SUGGESTED READINGScientists in the life sciences who are interested
in modelling should try to interest physicists (or
mathematicians with an open mind) in their problems. But
before committing to a given scientist, they should be able
to feel if the choice is good and therefore some personal
investment on the basics would be helpful. Two good
graduate level reminders of mathematics are the manuals
by Mathews and Walker ## CONCLUSION: GOALSTo understand is a noble and important goal, but it is not the only one. Scientists have a responsibility to help decide the best policies. The most obvious goal is to keep the climate and the biosphere compatible with life, and to make sure the planet can feed people decently. But it is also important to maintain some protected areas, firstly to conserve biodiversity, but also for purely “aesthetic” reasons, to keep some “virgin” areas as witness of what the Earth was when man was less omnipresent. To achieve these goals, purely predictive tools based on previously accumulated data are likely to err badly when out of the range of their database. Mechanistic models are better candidates, as they allow a deep understanding of the complexity of a natural mechanism (e.g., of an ecosystem), including its self-organization properties and its resilience to perturbations. Modelling is not an expensive activity, but it brings very great rewards, and it should be actively pursued by scientists in every country. Some inspirational reading is to be found in Arcadia
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