Background/Question/Methods   In many spatial models, a population invading an unoccupied environment will expand as an invasion wave of fixed shape moving at a constant speed. This invasion wave speed depends on both demography (i.e., on the rates of survival, development, reproduction, etc.) and on dispersal (i.e., on the probability distribution of distances dispersed by individuals at each stage of their life cycle). Because the wave speed integrates demography and dispersal into a single index of population spread, it plays a role analogous to that played by the population growth rate in demographic analysis. An important factor usually excluded from this idealized situation is environmental variation. Temporally varying environments have been analyzed in the special case of unstructured, scalar populations. The results, however, are fundamentally inapplicable to age- or stage-structured models, because multiplication by scalars is commutative, while multiplication by matrices is not. We have developed invasion models for structured populations in periodically and stochastically varying environments. Periodic models describe seasonal variation within a year or interannual variability that is (or can be thought of as) periodic. Stochastic models can be linked to explicitly stochastic models for environmental processes (e.g., fires, floods, or prey availability).

Results/Conclusions   In this presentation, we will demonstrate how such models, based on matrix integrodifference equations, are constructed. We will present explicit formulae for (a) the invasion wave speed and (b) the sensitivity and elasticity of the wave speed to both demographic and dispersal parameters.