Mathematical Models

It is not enough in biology to know what animals do and how they do it. It is also important to understand the adaptive significance of morphological and behavioural features, and hence why animals have evolved to look and behave the way they do. Evolution can be regarded as a process of improvement (or optimisation) as animals become better adapted to what they have to do in order to survive and reproduce in contemporary environments. So-called optimisation analysis is a powerful approach to the study of adaptation, and has been increasingly used to test hypotheses about bird migration (Alerstam & Lindström 1990, Alerstam & Hedenstrom 1998a). Models are used to predict behavioural and other patterns that 'should' be observed if individuals follow one strategy or another, and the predictions are compared to field observations or experimental findings to see how well they fit, and hence to infer the likely strategy being followed by the bird. Optimal behavioural decisions during migration may involve matters of habitat selection, flight speed and altitude, whether to fly now or later, whether by day or by night, by flapping or soaring flight, which direction to head, how much account to take of winds, and so on. The theory of bird flight yields quite specific predictions on the speed and altitude of flight and how it is expected to vary with wind or fuel loads, all of which can be tested with field data.

Although the combination of modelling and field observation comprises a potentially powerful method for studying migratory adaptations, such models are heavily dependent on the assumptions on which they are based. These assumptions may be unrealistic and in any case are dependent on current knowledge which may be inadequate. Such models are nonetheless important in directing research, through defining more precise questions and the types of data that need to be collected. So far in migration research, models have proved especially useful in understanding flight behaviour, patterns of fattening, the timing and duration of the individual flights and stopovers that comprise migration, and the responses of birds to wind conditions (see papers in Alerstam & Hedenstrom 1998b). At present, there are more models than critical tests of their assumptions and predictions, giving plenty of scope for further research. Like any other ideas, however, formal models (often couched in mathematical terms) must be continually tested against experiments and field observations. Progress is often most rapid when predictions fail or are not supported by new data, showing that seemingly plausible ideas are probably wrong.

Optimality models have another potential pitfall. Optimality does not require that all individuals in a population behave identically. It requires only that individuals make decisions that maximise their own fitness, including making the best of a bad job. Different individuals may therefore pursue different tactics, dependent on their own physical condition at the time, and on the prevailing environment as it affects them. Considerable variation in individual behaviour might then occur. The population might appear to be following no particular strategy, when in reality all individuals are behaving optimally for their own particular circumstances. One cannot interpret the significance of variability without information on the environmental conditions that affect the state and performance of the birds themselves at the time. Factors that influence individual variation are increasingly being incorporated into optimality modelling. In this book, I shall not dwell on the mathematical details of the various models (which are under continual revision), but with the physiological and ecological understanding that has emerged from their use.

As the study of migration has progressed over the years, rigorous statistical techniques have been developed to analyse the resulting data. Examples include the estimation of migration routes (Perdeck & Clason 1983), rates of movement (Nichols & Kaiser 1999), stopover durations and turnover rates of migrants at particular sites (Kaiser 1999), as well as the application of so-called circular statistics to the analysis of directional information (Batschelet 1981, Busse & Trocinska 1999).

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