This paper describes the use of linear deterministic models for examining the spread of population processes, discussing their advantages and limitations. Their main advantages are that their assumptions are relatively transparent and that they are easy to analyze, yet they generally give the same velocity as more complex linear stochastic and nonlinear deterministic models. Their simplicity, especially if we use the elegant reproduction and dispersal kernel formulation of Diekmann and van den Bosch et al., allows us greater freedom to choose a biologically realistic model and greatly facilitates examination of the dependence of conclusions on model components and of how these are incorporated into the model and fitted from data. This is illustrated by consideration of a range of examples, including both diffusion and dispersal models and by discussion of their application to both epidemic and population dynamic problems. A general limitation on fitting models results from the poor accuracy of most ecological data, especially on dispersal distances. Confirmation of a model is thus rarely as convincing as those cases where we can clearly reject one. We also need to be aware that linear models provide only an upper bound for the velocity of more realistic nonlinear stochastic models and are almost wholly inadequate when it comes to modeling more complex aspects such as the transition to endemicity and endemic patterns. These limitations are, however, to a great extent shared by linear stochastic and nonlinear deterministic models.
