Simple and complex models

Before we begin constructing our first models it is appropriate to pause and think about the rationale of model construction. The complexity of ecology and evolution provide both their fascination and frustration. We are faced with a myriad of species interacting with a variety of abiotic factors, both of which vary in time and space. How then can we begin to model these systems? There are two extremes of approach which have been described by various authors; for example, Maynard Smith's (1974) distinction between practical 'simulations' for particular cases and general 'models', May's (1973a) distinction, following Holling (1966), between detailed 'tactical' models and general strategic models and Levins' (1966) 'contradictory desiderata of generality, realism and precision'.

At the 'tactical' end of the spectrum we attempt to measure all the relevant factors and determine how they interact with the target system, such as a population. For example, in producing a model of change in plant numbers with time we might find that the plants are affected by 12 factors, such as summer rainfall, winter temperatures and levels of herbivory. This information is obtained through field observations and field or laboratory experiments. All the information is combined into a computer program, initial conditions are set (e.g. the number of plants at time 1), values for the different factors entered (e.g. the amount of summer rainfall) and the model run. The output of the model, in this case the number of plants at time t, is then revealed after different periods of time. This is a classic simulation exercise which has become feasible and easy to execute with high computer processing speeds and wide availability of appropriate software.

Now comes the tricky part. We have produced a realistic model in the sense that it mimics closely what we believe is happening in the field. However, we do not really know why it produces a certain answer. The model is intractable (and perhaps unpredictable) owing to its complexity. Tweaking a variable such as rainfall may radically change the output but we may not know why. In other words we have produced a black box which receives a set of variables and generates numbers that vary in time and space. One value of such a model is that it can speed up natural processes so that we do not have to wait 100 years to see how the plant population will (possibly) change, assuming factors remain the same or change in a predictable manner. To get closer to the mechanism(s) in these types of model we have two options. The first is to alter the variables systematically and see how the output responds. This is perhaps best undertaken after the second option, which is to strip the model down to its statistically significant components. You will recall from Chapter 1 that one feature of multiple regression is the removal of non-significant explanatory variables. This will include removing explanatory variables that are correlated. Multiple regression is one example of a set of statistical methods which allow the removal of non-significant terms, resulting in the simplest realistic model (often referred to as the minimal adequate model, especially in connection with particular statistical applications). These methods are consistent with the guiding principle of parsimony which states that the simplest explanation is the best one (Occam's razor). This principle is relevant to all branches of science. The identification of a common third and hidden variable in the power functions in Chapter 1 follows the principle of parsimony as it replaces two variables by one. In evolutionary biology parsimony has been of fundamental importance in the construction of phylogenetic trees.

From the 'strategic' end of the spectrum we can create a model which is so simple that it is known to be unrealistic. What is the value in such an approach? Here the objective is rather different to the simplest realistic model generated from the tactical end. We are using mathematical modelling as a way of formalizing generalizations about the study system. The model is not derived out of consideration of one particular example. It can be argued that strategic models are the most important types of model as they lie at the core of realistic models. If we do not understand the mode of operation of strategic models then we can never understand why the particular realistic models do what they do. From a mathematical perspective, strategic models are often designed so that their properties can be revealed through analytical solutions of the underlying equations. For example, the stability boundaries of some simple mathematical models can be determined by manipulation of the equations whereas more complex models cannot be solved in this manner. Examples of these types of solution will be given later in the book.

In the light of this discussion we will begin by exploring the properties of some simple strategic models. In fact the following could be argued as strategic models arising from realistic considerations, encompassing the best of the strategic and tactical approaches.

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