As you may be starting to appreciate, there are few areas of science which cannot benefit from an understanding of mathematical models or cannot be couched in mathematical terms. The introductory methods covered in this book should allow you to start developing your own models and provide an insight into the methods commonly used by a wide variety of practitioners. Hopefully you will begin to appreciate the wide range of applications, including biological control, sustainable harvesting and conservation management.
An important distinction in mathematical modelling, and one which informs the organization of the next two chapters, is between deterministic and stochastic processes. In a deterministic world everything should be predictable. For example, if population dynamics are deterministic we should be able to predict the population size at time t given a knowledge of the processes (described by mathematical equations) underlying the dynamics. Simple models with deterministic dynamics are the subject of Chapter 2. This notion of the deterministic world is undermined in two distinct ways. First, deterministic processes do not necessarily lead to predictable outcomes (as we shall see in Chapter 5) and second that stochastic or random events may be as important in ecological and evolutionary dynamics, as we will discover in Chapter 3. In most cases a combination of stochastic and deterministic modelling is the best way to proceed. To use the regression analogy, we need to identify and quantify the deterministic signal (the variation due to the regression) and we need to find ways of modelling the unexplained variance, which may be the result of extrinsic random events and/or sampling error. We also need to be aware that processes which are stochastic at one temporal or spatial scale may be much more predictable at larger scales. So, while it is difficult to predict the occurrence of storms from day to day, we may be much more certain about their occurrence and even their strength from month to month.
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