Periodicity versus Chaos in One-Dimensional Dynamics

We survey recent results in one-dimensional dynamics, and as an application, we derive rigorous basic dynamical facts for two standard models in population dynamics, the Ricker and the Hassell families. We also informally discuss the concept of chaos in the context of one-dimensional discrete time models. First we use the model case of the quadratic family for an informal exposition and then we review precise generic results, before we turn to the population models. Our focus is on typical asymptotic behavior, seen for most initial conditions and for large sets of maps. Parameter sets corresponding to different types of attractors are described. In particular it is shown that maps with strong chaotic properties appear with positive frequency in parameter space in our population models. Natural measures (asymptotic distributions) and their stability properties are considered.