## Mathematical models introduction

The desire to formulate general rules in ecology often finds its expression in the construction of mathematical or graphical models. It may seem surprising that those interested in the natural living world should spend time reconstructing it in an artificial mathematical form; but there are several good reasons why this should be done. The first is that models can crystallize, or at least bring together in terms of a few parameters, the important, shared properties of a wealth of unique examples. This simply makes it easier for ecologists to think about the problem or process under consideration, by forcing us to try to extract the essentials from complex systems. Thus, a model can provide a 'common language' in which each unique example can be expressed; and if each can be expressed in a common language, then their properties relative to one another, and relative perhaps to some ideal standard, will be more apparent.

These ideas are more familiar, perhaps, in other contexts. Newton never laid hands on a perfectly frictionless body, and Boyle never saw an ideal gas - other than in their imaginations - but Newton's Laws of Motion and Boyle's Law have been of immeasurable value to us for centuries.

Perhaps more importantly, however, models can actually shed light on the real world that they mimic. Specific examples below will make this apparent. Models can, as we shall see, exhibit properties that the system being modeled had not previously been known to possess. More commonly, models make it clear how the behavior of a population, for example, depends on the properties of the individuals that comprise it. That is, models allow us to see the likely consequences of any assumptions that we choose to make - 'If it were the case that only juveniles migrate, what would this do to the dynamics of their populations?' - and so on. Models can do this because mathematical methods are designed precisely to allow a set of assumptions to be followed through to their natural conclusions. As a consequence, models often suggest what would be the most profitable experiments to carry out or observations to make - 'Since juvenile migration rates appear to be so important, these should be measured in each of our study populations'.

These reasons for constructing models are also criteria by which any model should be judged. Indeed, a model is only useful (i.e. worth constructing) if it does perform one or more of these functions. Of course, in order to perform them a model must adequately describe real situations and real sets of data, and this 'ability to describe' or 'ability to mimic' is itself a further criterion by which a model can be judged. However, the crucial word is 'adequate'. The only perfect description of the real world is the real world itself. A model is an adequate description, ultimately, as long as it performs a useful function.

In the present case, some simple models of intraspecific competition will be described. They will be built up from a very elementary starting point, and their properties (i.e. their ability to satisfy the criteria described above) will then be examined. Initially, a model will be constructed for a population with discrete breeding seasons.

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