Prof. Dr. Odo Diekmann, Utrecht University A delay equation is a rule for extending a function of time towards the future, on the basis of the known past. Renewal Equations prescribe the current value, while Delay Differential Equations prescribe the derivative of the current value. With a delay equation one can associate a dynamical system by translation along the extended function. I will illustrate by way of examples how such equations arise in the description of the dynamics of structured populations and sketch the available theory, while making a plea for the development of numerical bifurcation tools. The lecture is based on joint work with Mats Gyllenberg, Hans Metz and many others. The spectral radius of a positive operator: eigenvectors, bounds, and approximation by power methods Horst R. Thieme, Arizona State University In structured population models, the basic reproduction number often is the spectral radius of an appropriate positive linear operator on an ordered Banach space. This operator is called next generation operator in case a biological interpretation is available. Since a closed expression for its spectral radius can only be obtained in special cases, there is renewed interest in the approximation and estimation of the spectral radius. Quite a few results are available in the operator theory and computational/numerical literature. It is one of the purposes of this talk to review some of these and perhaps give them a new twist. Motivated by two-sex population models, we extend our study to homogeneous increasing maps on cones. We present conditions for convergence of iterates to balanced geometric growth (strong ergodicity) and for convergence of power methods to the cone spectral radius and the associated eigenvector.