ABSTRACT
Numerical simulation provides a very useful tool for the investigation of population models. As such, it complements analytical calculations. Analysis can provide formulae for the statistics of the population, whereas simulation can give actual examples of populations and their evolutions. This provides additional insight into the models; numerical techniques have been employed for this purpose in the present book. In the language of statistics, we can say that the analytical approach gives information about the behaviour of ensembles of populations, whereas numerical calculations generate realisations of the populations. However, numerical analysis can also give estimates of ensemble properties, by averaging data from many realisations. The results are only estimates because only a finite number of realisations can be averaged, whereas a true ensemble average includes contributions from every possible state of the population; thus the estimates are subject to errors and a certain amount of insight is required to understand these errors. Numerical estimates of ensemble properties can be useful for checking the results of analytical calculations or for providing results in regimes where such calculations are intractable. Conversely, some models, such as the stable processes discussed in chapter 7, may present difficulties for numerical analysis, because some ensemble properties are determined by very rare, large events.
