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Outline of Modelling Techniques and their Interrelationships |
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3.3.1 Functional relationships |
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3.3.2 Matrix models |
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3.3.3 Statistical models |
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3.3.4 Multivariate models |
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3.3.5 Mathematical programming |
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3.3.6 Game theory models |
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3.3.7 Catastrophe theory |
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In this Handbook, the word 'model' is taken to mean a formal expression of the essential elements of some problem in either physical or mathematical terms. In past scientific work, much of the emphasis in scientific explanation has been on the use of physical analogues of natural phenomena, and there is still a need for physical analogues in the study of biological and environmental processes. More generally, however, the models with which we are concerned here will be mathematical and essentially abstract.
We have explicitly excluded 'word models' or purely verbal representations of problems from further consideration in this Handbook. It is true, of course, that our first recognition of any problem is likely to be expressed in words, and there is much to be gained by seeking the most precise description that can be made of any problem with which we are concerned. It is surprising how often even four or five people closely concerned with a problem will disagree with each other's descriptions of the same ecological system, and disagreement on the particular elements of the systems which contribute, directly or indirectly, to a problem of practical concern is even more likely. For the larger groups of scientists, resource managers and administrators likely to be concerned with problems of dynamic change, the disagreement may be both striking and difficult to resolve. There is, therefore, all the more reason to spend some time in an attempt to find an agreed description, even if that description contains some passages, expressed as alternatives, for which no agreement can be reached.
Our reason for concentrating on the use of mathematical expressions as opposed to verbal expressions and physical analogues lies in the ability of mathematics to provide a symbolic logic which is capable of describing ideas, and particularly relationships, of very great complexity, while, at the same time, retaining a simplicity and parsimony of statement. The whole basis of mathematical notation rests on this economical description of relationships as a symbolic logic, and such description is 'formal' in the sense that it enables predictive statements to be derived from the relationships. Without the ability to predict the results of changes in one or more elements in the relationship, we could not regard our statements as belonging to science rather than to metaphysics or to literature.
The use of a mathematical notation in the modelling of complex systems is, therefore, an attempt to provide a representational symbolic logic which simplifies, but does not markedly distort, the underlying relationships. The use of symbolic logic, because it is essentially a simplification, necessarily gives an imperfect representation of reality, and must therefore be regarded as a caricature. Nevertheless, the various mathematical rules for manipulating the relationships enable predictions to be derived of changes which may be expected to occur with time in ecological systems as various component values of these systems are changed. These predictions, in turn, enable comparisons to be made between model systems and the real systems which they are intended to represent, and, in this way, to test the adequacy of the model against observations and data derived from the real world. This 'appeal to nature' is an essential part of the scientific method. Indeed, manipulation of the model system may itself suggest the experiments which are necessary to test the adequacy of the system.
The advantages of formal mathematical expressions as models are:
they are precise and abstract;
they transfer information in a logical way; and
they act as an unambiguous medium of communication.
They are precise because they enable predictions to be made in such a way that these predictions can be checked against reality by experiment or by survey. They are abstract because the symbolic logic of mathematics extracts those elements, and only those elements, which are important to the deductive logic of the argument, thus eliminating all the extraneous meanings which may be attached to the words. Mathematical models transfer information from the whole body of knowledge of the behaviour of interrelationships to the particular problem being investigated, so that logically dependent arguments are derived without the necessity of repeating all the past research. Mathematical models also provide a valuable means of communication because of the unambiguity of the symbolic logic employed in mathematics, a medium of communication which is largely unaffected by the normal barriers of human language.
The disadvantage of mathematical models lie in the apparent complexity of the symbolic logic, at least to the non-mathematician. In part, this is a necessary complexity __ if the problem under investigation is complex, it is likely, but not certain, that the mathematics needed to describe the problem will also be complex. There is also a certain opaqueness of mathematics, and the difficulty that many people have in translating from mathematical results to real life is not confined to non-mathematicians. It is, therefore, always important to ensure that the results of mathematical analysis are correctly interpreted and to translate solutions from mathematical formulae into everyday language.
Perhaps the greatest disadvantage of mathematical models, however, is the distortion that may be introduced into the solution of a problem by inflexible insistence on a particular model, even when it does not really fit the facts. The pursuit of mathematical models is sometimes intoxicating, to the extent that it is relatively easy for scientists to abandon the real world and to indulge in the use of mathematical languages for what are essentially abstract art forms. It is for this reason that we will insist, in this Handbook, that models are fully embedded in a broad framework of systems analysis.
In the sense in which we shall use the term in this Handbook, 'systems analysis' is the orderly and logical organization of data and information into models, followed by the rigorous testing and exploration of these models necessary for their validation and improvement. Systems analysis provides a framework of thought designed to help decision-makers to choose a desirable course of action, or to predict the outcome of one or more courses of action that seem desirable to those who have to make decisions. In particularly favourable cases, the course of action that is indicated by the systems analysis will be the 'best' choice in some specified or defined way.
The aim of the broad framework of systems analysis outlined in the next chapter is to promote good decision-making in practical applications, and, in our case, in the ecology of dynamic systems. The framework is intended to focus and to force hard thinking about complex, and usually large, problems not amenable to solution by simpler methods of investigation, for example by direct experimentation or by survey. The special contribution of systems analysis lies: (i) in the identification of unanticipated factors and interactions that may subsequently prove to be important, (ii) in the forcing of modifications to experimental and survey procedures to include these factors and interactions, and (iii) in illuminating critical weaknesses in hypotheses and assumptions. Just as the scientific method, with its insistence on the testing of hypotheses through practical experiments and rigorous sampling procedures, provides the essential tools for advances in our knowledge of the phsyical world, systems analysis welds these tools into a flexible, but rigorous, exploration of complex phenomena.
The inherent complexity of ecological relationships, the characteristic variability of living organisms and the apparently unpredictable effects of deliberate modification of ecosystems by man necessitate an orderly and logical organization of ecologial research which goes beyond the sequential application of tests of simple hypotheses, although the 'appeal to nature' invoked by the experimental method necessarily remains at the heart of this organization. Applied systems analysis provides one possible format for this logical organization, a format in which the experimentation is embedded in a conscious attempt to model the system so that the complexity and the variability are retained in a form in which they are amenable to analysis. A further reason for the use of systems analysis in ecological research lies in the relatively long timescales which are required for that research. It is, therefore, necessary to ensure the greatest possible advance from each stage of the experimentation, and the models of systems analysis provide the necessary framework for such advances. Furthermore, the present state of ecology as a science, with its extremely scattered research effort over a wide field, urgently needs a unifying concept. Not only is there a marked incompatibility of the many existing theories, but the weakness of the assumption behind these theories is largely unexplored, partly because the assumptions themselves have never been stated. Systems analysis can, therefore, be used as a filter of existing ideas. Theories which have been shown to be incompatible can be tested as alternative hypotheses, and the systems analysis itself will frequently suggest the critical experiments necessary to discriminate between these hypotheses.
While some of the general properties of mathematical models have been touched upon above, an experienced systems analyst can recognize broad 'families' of models, in much the same way that an experienced botanist is often able to identify the genus to which the plant belongs, even when he does not know the species. It may, therefore, be useful to review briefly some of the main families which can be recognized among mathematical models. The list is far from being exhaustive, and the categories are also not mutually exclusive. The classification is, however, probably sufficient to provide examples of the basic requirements of models applied to practical problems.
Many ecological models are based on studies of systems dynamics which are themselves based on servo-mechanism theory, coupled with the use of high-speed digital computers to solve large numbers of equations in a short time. These equations are more or less complex mathematical descriptions of the operation of the system being simulated, and are in the form of expressions for levels .of various types which change at rates controlled by decision functions. The level equations represent accumulations within the ecological system of such variables as weight, numbers of organisms and energy, and the rate equations govern the change of these levels with time. The decision functions represent the policies or rules, explicit or implicit, which are assumed to control the operation of the system.
The popularity of dynamic models of this kind arises from the flexibility of the models to describe systems operations, including non-linear responses of components to controlling variables and both positive and negative feedback. However, this flexibility has its disadvantages. It is, in any case, usually impossible to include equations for all the components of a system, as, even with modern computers, the simulation rapidly becomes too complex. It is, therefore, necessary to obtain an abstraction based on judgement and on assumptions as to which of the many components are those which control the operation of a system.
The application of systems dynamics in modelling involves three principal steps. First, it is necessary to identify the dynamic behaviour of the system of interest, and to formulate hypotheses about the interactions that create the behaviour. Second, a computer simulation model must be created in such a way that it replicates the essential elements of the behaviour and interactions identified as essential to the system. Third, when it has been confirmed that the behaviour of the model is sufficiently close to that of the real system, the model can be used to understand the cause of observed changes in the real system, and to suggest experiments to be carried out in the evaluation of potential courses of action.
Systems dynamic models have an intuitive appeal. The formulation of the models allows for considerable freedom from constraints and assumptions, and for the introduction of the non-linearity and feedback which are apparently characteristic of many ecological systems. The ecologist is thus able to mirror or mimic the behaviour of the system as he understands it, and gain some useful insight into the behaviour of the system as a result of changes in the parameters and driving variables. The power of computers to make large numbers of exact but small computations also enables the ecologist to replace the analytical techniques of integration by the less accurate, but easier, methods of difference equations. Furthermore, even when the values of parameters are unknown, relatively simple techniques exist to provide approximations for these parameters by sequential estimates, or even to use interpolations from tabulated functions. In particularly favourable cases, it may even be possible to test various hypotheses of parameters or functions.
The lack of a formal structure for such models and the freedom from constraints can, however, also be disadvantageous. For one thing, the behaviour of even quite simple dynamic models may be very difficult to predict, and it is easy to construct models whose behaviour, even within the practical limits of the input variables, is unstable or inconsistent with reality. Even more difficult, determination of the way in which such systems will behave frequently requires extensive and sophisticated experimentation. It is certainly always necessary to test the behaviour of such model in relation to the interaction of changes of two or more input variables, and seldom, if ever, sufficient to test the responses to changes in one variable at a time.
Further discussion of dynamic models is given in Chapter 5. To summarize, however, dynamic models may well be helpful in the early stages of the systems analysis of a complex and dynamic ecological problem, by concentrating attention on the basic relationships underlying the system, and by defining the variables and sub-systems that the investigator believes to be critical. In the later stages in the analysis, it will often be preferable to switch the main effort of analysis to one of the other families of models. It is precisely for this reason that systems analysis explicitly defines a phase of developing alternative solutions to the problem.
The family of dynamic models described in the previous section offers almost complete freedom to the investigator in the expression of those elements considered to be essential to the understanding of the underlying relationships between those variables and entities that have been identified in the description of the system. The models usually strive for 'reality' __ a recognizable analogy between the mathematics and the physical, chemical, or biological processes __ sometimes at the expense of mathematical elegance or convenience. The price paid for the 'reality' is frequently a necessity to multiply entities to account for relatively small variations in the behaviour of the system, or some difficulty in deriving unbiased or valid estimates of the model parameters.
Matrix models represent one family of models in which 'reality' is sacrificed to some extent in order to obtain the advantages of the particular mathematical properties of the formulation. The deductive logic of pure mathematics then enables the modeller to examine the consequences of his assumptions without the need for time-consuming 'experimentation' on the model, although computers may still be required for some of the computations.
Some of the most elegant of these matrix models are represented by the Leslie deterministic models predicting the future age structure of a population of animals from the present known age structure and assumed rates of survival and fecundity. Predator-prey systems, which sometimes show marked oscillations, can also be encompassed by matrix models, by a relatively simple exploitation of techniques for relating population size and survival. Seasonal and random environmental changes and the effects of time lags may similarly be incorporated, though the models necessarily become increasingly complex in formulation. Dynamic processes such as the cycling of nutrients and the flow of energy in ecosystems can also be modelled by the use of matrices, exploiting the natural compartmentation of the ecosystem into its species composition or into its trophic levels. Losses from the ecosystem are assumed to be the difference between the input and the sum of output from, and storage in, anyone compartment.
Although matrix calculations are sometimes extensive, especially in matrix inversion and in the calculation of eigenvalues and eigenvectors, and will often require the use of computers, they are usually less difficult to program than those involved in dynamic models. Furthermore, the properties of the basic matrices of the models enable the modeller to exploit the deductive logic of pure mathematics. Matrix models, therefore, represent an important and neglected family of models in systems analysis. A particular type of matrix model which is especially useful in the modelling of dynamic change in ecological systems is that of the Markov models, in which the basic format is of a matrix of entries expressing and probabilities of the transition from one state to another at specified intervals. These models will be considered in further detail in Chapter 6.
The families of models so far considered have been mainly deterministic. That is to say, from a given starting point, the outcome of the model's response is necessarily the same and is predicted by the mathematical relationships incorporated in the model. Such models are necessarily mathematical analogues of physical processes in which there is a one-to-one correspondence between cause and effect. There is, however, a later development of mathematics which enables relationships to be expressed in terms of probabilities, and in which the outcome of a model's response is not certain. Models which incorporate probabilities are known as stochastic models, and such models are particularly valuable in simulating the variability and complexity of ecological systems.
Statistical models of this kind include the models of spatial patterns of organisms, the analysis of variance, multiple regression analysis, and the Markov models mentioned in the last section. Although, apart from the Markov models, we will not be primarily concerned with statistical models in this Handbook, some mention of their application will be made in the other chapters, and particularly in Chapter 7.
A particular class of statistical models which has special relevance to the modelling of dynamic change in ecosystems is that of multivariate models, where it is necessary to consider changes between many variables, or variates. A variate is a quantity which may take anyone of the values of a specified set with a specified relative frequency or probability. Such variates are sometimes also known as random variables and are to be regarded as defined not merely by a set of permissible values like any ordinary mathematical variable, but also by an associated frequency or probability function expressing how often those values appear in the situation under discussion. There are many situations in ecology and other applications of systems analysis where models have to capture the behaviour of more than one variate. These models are known collectively as 'multivariate' and are related to techniques known collectively as 'multivariate analysis' __ an expression which is used rather loosely to denote the analysis of data which are multivariate, in the sense that each individual under investigation bears the values of p variates.
Broadly, these models may be divided into two main categories:
those in which some variates are used to predict others; and
those in which all the variates are of the same kind, and no attempt is made to predict one set from the other.
For the latter, which may be broadly described as descriptive models, there is a further subdivision into those models in which all the inputs are quantitative, and those models in which at least some of the inputs are qualitative rather than quantitative. Predictive models, on the other hand, may first be subdivided according to the number of variates predicted, and then by whether or not all the predictors are quantitative. The use of such models in the investigation of dynamic change in ecosystems is described in Chapter 7.
The term 'mathematical programming' describes a series of models whose aim is to find the maximum or minimum of some mathematical expression or function by setting values to certain variables which may be altered within defined limits. The underlying mathematics of these models was developed during the early application of mathematical techniques to practical problems in what has now come to be known as operational research. The simplest of these problems, in which the objective function and the constraints are all linear functions, can be solved relatively easily by standard methods. In essence, any inequalities in the constraints are first removed by introducing some additional 'slack' variables. Any feasible solution to the problem is then sought and, once such a solution has been found, iterative attempts are made to improve the solution, i.e. to move it closer to the defined optimum of the objective function by making small changes in the values of the variables. This iterative procedure continues until no further improvement can be made. Non-linearity in either the objective function or the constraints, or both, introduces disproportionate levels of difficulty in finding appropriate solutions. So, too, do problem formulations which impose limitations on the size of the lumps in which units of some particular resource can be allocated. There is, nevertheless, a well-developed theory of non-linear programming to cope with such problems.
As a further extension, some large optimization problems can be reformulated as a series of smaller problems, arranged in sequences of time or space, or both. A reformulation of this kind is frequently desirable in order to reduce the computational effort of finding a solution, although care has to be taken to ensure that the sum of the optimal solutions of the sub-problems approaches the optimum solution of the whole problem. Mathematical programming models are not widely used in the modelling of dynamic change in ecosystems and are not, therefore, further discussed in this Handbook.
Closely related to mathematical programming models are the models which are based on the theory of games. The simplest of these models is known as the two-person, zero-sum game. Such games are characterized by having two sets of interests represented, one of which may be nature or some external force, and by being 'closed' in the sense that what one player loses in the game the other must win. The theory can, however, be extended to many-person, non-zero-sum games. Such models represent an interesting, and so far little explored, alternative approach to the solution of strategic problems. The extension to the more complex non-zero-sum games and to many-person games, in which coalitions can be formed between the players, represents a field of research which deserves increased attention, particularly in ecological research related to the assessment of environmental impact and environmental planning. Because of a lack of practical examples of application of such models to the modelling of dynamic change in ecosystems, they will not be further discussed in this Handbook.
The theory of catastrophes is an elegant development of mathematical topology applied to systems which have four basic properties, namely bimodality, discontinuity, hysteresis and divergence. Catastrophe theory models have attracted much interest and attention since they were first proposed in 1970. The models have considerable intellectual and visual appeal, but are not easy to apply in highly multivariate situations. There are also serious difficulties to be overcome in estimating the parameters of the model from ecological data. Again, although we are likely to see wider use of such models in the study of dynamic change in ecosystems, further discussion of such models will not be attempted in this Handbook.
It is clear that the list of families of models described above is far from being exhaustive, and that the categories are also not mutually exclusive. Thus, Markov models belong both to the family of matrix models and to the family of statistical models. Furthermore, some basic statistical models are frequently essential to the development of dynamic models.
The distinction between deterministic and stochastic models is of particular importance. With a deterministic model, one will always arrive at the same predictions for given starting values, and for given values of the coefficients or parameters of the model. If, on the other hand, a stochastic model is used as a basis for simulation, the outcome of the simulation will not always be the same, even when the parameters and starting values are the same. The random elements in the model ensure variability, and the aim of such models is to mirror the variability found in living organisms and in ecological systems. As with experiments on the organisms themselves, it will usually be necessary to make repeated trials of the simulation in order to determine the ways in which the system will respond to various changes.
A further important distinction is related to the dimensionality of models, i.e. to the number of independent dimensions or variables that are included in the model system. Many variables will be inter-correlated and the actual number of independent dimensions will, therefore, be smaller than the total number of variables. One of the most valuable characteristics of the multivariate models described in Chapter 7 of this Handbook is that they help to determine the true dimensionality of the model system, and to select the most critical and useful variables upon which to base the simulation.
Again, while we have classified broad groups of models into families according to their mathematical characteristics, we could also classify models according to their purposes. Maynard-Smith (1974) makes a distinction between 'models' and 'simulations'. He regards a mathematical description with a practical purpose, which includes as much relevant detail as possible, as a 'simulation', and restricts the use of the word 'model' to descriptions of general ideas which include a minimum of detail. This is not a distinction which will be maintained in this Handbook.
Many of the models that we may wish to construct of dynamic change in ecological systems may be regarded as descriptive models. Their aim is to provide as good a description as possible of the underlying processes or relationships on which dynamic change depends. Such models may help to weld together widely disparate theories about ecological systems, or, alternatively, they may help to show the incompatibility of commonly held beliefs or theories. Once formulated, the descriptive model will be used to obtain a further understanding of the way in which a single organism, a community of organisms, or a whole ecosystem will respond to various changes, natural or induced.
Alternatively, a model may be described primarily for the purpose of prediction. Such predictive models may pay relatively little attention to the physics, chemistry or biology of the underlying processes, but will be regarded as efficient only if they enable predictions of future states of the system to be made with a known degree of accuracy. Predictive models may be derived as the operational versions of descriptive models, the improved knowledge of the underlying processes being used to refine the predictive capability of the mathematical expressions. It may, nevertheless, be possible to develop predictive models directly, sometimes with drastic simplification of the mathematical assumptions about ecological processes. Efficient descriptions and efficient predictions are not necessarily closely related.
A third class of models may be distinguished as decision models. The aim of such models is neither to provide a description of the ecological system nor to predict the future state of such a system, but, instead, to guide practical decisions about the management or treatment of the system. In a sense, of course, both descriptive and predictive models can be used to guide practical decisions. Decision models are, however, specially formulated so as to provide such guidance more directly, and, in addition, by showing the consequences of particular choices about the management of an ecological system, they help to indicate a management strategy which is 'best' in some predefined way. Decision models are not usually derived from descriptive or predictive models, but are developed from families of models which have distinctive mathematical properties, like mathematical programming.
As this Handbook cannot be a comprehensive guide to the whole set of model families, a limited number of models have been chosen as being of greatest value in the modelling of dynamic change in ecosystems. Three main classes of models are described in detail, namely dynamic or functional models, Markov models, and multivariate models. For all three classes, there is now sufficient experience of their application to the modelling of dynamic change to enable an assessment to be made of their usefulness, and of the difficulties of constructing models from the kinds of data likely to be available. Future revisions of this Handbook will almost certainly include other classes of models as expertise is developed in their use and construction.
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